Rigorous three-dimensional relativistic equation for quark-antiquark bound states at finite temperature derived from the thermal QCD formulated in the coherent-state representation

A rigorous three-dimensional relativistic equation for quark-antiquark bound states at finite temperature is derived from the thermal QCD generating functional which is formulated in the coherent-state representation. The generating functional is derived newly and given a correct path-integral expression. The perturbative expansion of the generating functional is specifically given by means of the stationary-phase method. Especially, the interaction kernel in the three-dimensional equation is derived by virtue of the equations of motion satisfied by some quark-antiquark Green functions and given in a closed form which is expressed in terms of only a few types of Green functions. This kernel is much suitable to use for exploring the deconfinement of quarks. To demonstrate the applicability of the equation derived, the one-gluon exchange kernel is derived and described in detail

Massive Gauge Field Theory Without Higgs Mechanism I. .Quantization

According to the conventional concept of the gauge field theory, the local gauge invariance excludes the possibility of giving a mass to the gauge boson without resorting to the Higgs mechanism because the Lagrangian constructed by adding a mass term to the Yang-Mills Lagrangian is not only gauge-non-invariant, but also unrenormalizable. On the contrary, we argue that the principle of gauge invariance actually allows a mass term to enter the Lagrangian if the Lorentz constraint condition is taken into account at the same time. The Lorentz condition, which implies vanishing of the unphysical longitudinal field, defines a gauge-invariant physical space for the massive gauge field. The quantum massive gauge field theory without Higgs mechanism may well be established by using a BRST-invariant action which is constructed by the Lagrange undetermined multiplier procedure of incorporating the Lorentz condition and another condition constraining the gauge group into the original massive Yang-Mills action. The quantum theory established in this way shows good renormalizability.Comment: 34 pages, latex, 3 figure

059<p type="texpara" tag="Body Text" et="f_0" bin="clone" >Massive Gauge Field Theory Without Higgs Mechanism Massive Gauge Field Theory Without Higgs Mechanism III. Proof of Renormalizability

~It is shown that the quantum massive non-Abelian field theory established in the former papers is renormalizable. This conclusion is achieved with the aid of the Ward-Takahashi identities satisfied by the generating functionals which were derived in the preceding paper based on the BRST-symmetry of the theory. By the use of the Ward-Takahashi identity, it is proved that the divergences occurring in the perturbative calculations for the massive gauge field theory can be eliminated by introducing a finite number of counterterms in the effective action. As a result of the proof, it is found that the renormalization constants for the massive gauge field theory comply with the same Slavnov-Taylor identity as that for the massless gauge field theory. The latter identity is re-derived from the Ward-Takahashi identities satisfied by the gluon proper vertices and their renormalization.Comment: 25 pages, latex, no figure

Lorentz-Covariant Quantization of Massive Non-Abelian Gauge Fields in The Hamiltonian Path-Integral Formalism

The massive non-Abelian gauge fields are quantized Lorentz-covariantly in the Hamiltonian path-integral formalism. In the quantization, the Lorentz condition, as a necessary constraint, is introduced initially and incorporated into the massive Yang-Mills Lagrangian by the Lagrange multiplier method so as to make each temporal component of a vector potential to have a canonically conjugate counterpart. The result of this quantization is confirmed by the quantization performed in the Lagrangian path-integral formalism by applying the Lagrange multiplier method which is shown to be equivalent to the Faddeev-Popov approach

Massive Gauge Field Theory Without Higgs Mechanism II. Ward-Takahashi Identities and Proof of Unitarity

In our previously published papers, it was argued that a massive non-Abelian gauge field theory in which all gauge fields have the same mass can well be set up on the gauge-invariance principle. The quantization of the fields was performed by different methods. In this paper, It is proved that the quantum theory is invariant with respect to a kind of BRST-transformations. From the BRST-invariance of the theory, the Ward-Takahashi identities satisfied by the generating functionals of full Green's functions, connected Green's functions and proper vertex functions are successively derived. As an application of the above Ward-Takahashi identity, the Ward-Takahashi identity obeyed by the massive gauge boson propagator is derived and the renormalization of the propagator is discussed. Furthermore, based on the Ward-Takahashi identity, it is exactly proved that the S-matrix elements given by the quantum theory are gauge-independent and hence unitary.Comment: 18 pages, latex, no figure

Quantization of The Electroweak Theory in The Hamiltonian Path-Integral Formalism

The quantization of the SU(2)$\times$U(1) gauge-symmetric electroweak theory is performed in the Hamiltonian path-integral formalism. In this quantization, we start from the Lagrangian given in the unitary gauge in which the unphysical Goldstone fields are absent, but the unphysical longitudinal components of the gauge fields still exist. In order to eliminate the longitudinal components, it is necessary to introduce the Lorentz gauge conditions as constraints. These constraints may be incorporated into the Lagrangian by the Lagrange undetermined multiplier method. In this way, it is found that every component of a four-dimensional vector potential has a conjugate counterpart. Thus, a Lorentz-covariant quantization in the Hamiltonian path-integral formalism can be well accomplished and leads to a result which is the same as given by the Faddeev-Popov approach of quantization.Comment: 9 pages, no figure

Lorentz-Covariant Quantization of Massless Non-Abelian Gauge Fields in The Hamiltonian Path-Integral Formalism

The Lorentz-covariant quantization performed in the Hamiltonian path-integral formalism for massless non-Abelian gauge fields has been achieved. In this quantization, the Lorentz condition, as a constraint, must be introduced initially and incorporated into the Yang-Mills Lagrangian by the Lagrange undetermined multiplier method. In this way, it is found that all Lorentz components of a vector potential have thier corresponding conjugate canonical variables. This fact allows us to define Lorentz-invariant poisson brackets and carry out the quantization in a Lorent-covariant manner. Key words: Non-Abelian gauge field, quantization, Hamiltonian path-integral formalism, Lorentz covariance.Comment: 11 pages no figure

Massive Gauge Field Theory Without Higgs Mechanism IV. Illustration of Unitarsity

To illustrate the unitarity of the massive gauge field theory described in the foregoing papers, we calculate the scattering amplitudes up to the fourth order of perturbation by the optical theorem and the Landau-Cutkosky rule. In the calculations, it is shown that for a given process, if all the diagrams are taken into account, the contributions arising from the unphysical intermediate states included in the longitudinal part of the gauge boson propagator and in the ghost particle propagator are completely cancelled out with each other in the S-matrix elements. Therefore, the unitarity of the S-matrix is perfectly ensured.Comment: 30 pages, latex, 9 figure

Some Extensions of Probabilistic Logic

In [12], Nilsson proposed the probabilistic logic in which the truth values of logical propositions are probability values between 0 and 1. It is applicable to any logical system for which the consistency of a finite set of propositions can be established. The probabilistic inference scheme reduces to the ordinary logical inference when the probabilities of all propositions are either 0 or 1. This logic has the same limitations of other probabilistic reasoning systems of the Bayesian approach. For common sense reasoning, consistency is not a very natural assumption. We have some well known examples: {Dick is a Quaker, Quakers are pacifists, Republicans are not pacifists, Dick is a Republican}and {Tweety is a bird, birds can fly, Tweety is a penguin}. In this paper, we shall propose some extensions of the probabilistic logic. In the second section, we shall consider the space of all interpretations, consistent or not. In terms of frames of discernment, the basic probability assignment (bpa) and belief function can be defined. Dempster's combination rule is applicable. This extension of probabilistic logic is called the evidential logic in [ 1]. For each proposition s, its belief function is represented by an interval [Spt(s), Pls(s)]. When all such intervals collapse to single points, the evidential logic reduces to probabilistic logic (in the generalized version of not necessarily consistent interpretations). Certainly, we get Nilsson's probabilistic logic by further restricting to consistent interpretations. In the third section, we shall give a probabilistic interpretation of probabilistic logic in terms of multi-dimensional random variables. This interpretation brings the probabilistic logic into the framework of probability theory. Let us consider a finite set S = {sl, s2, ..., Sn) of logical propositions. Each proposition may have true or false values; and may be considered as a random variable. We have a probability distribution for each proposition. The e-dimensional random variable (sl,..., Sn) may take values in the space of all interpretations of 2n binary vectors. We may compute absolute (marginal), conditional and joint probability distributions. It turns out that the permissible probabilistic interpretation vector of Nilsson [12] consists of the joint probabilities of S. Inconsistent interpretations will not appear, by setting their joint probabilities to be zeros. By summing appropriate joint probabilities, we get probabilities of individual propositions or subsets of propositions. Since the Bayes formula and other techniques are valid for e-dimensional random variables, the probabilistic logic is actually very close to the Bayesian inference schemes. In the last section, we shall consider a relaxation scheme for probabilistic logic. In this system, not only new evidences will update the belief measures of a collection of propositions, but also constraint satisfaction among these propositions in the relational network will revise these measures. This mechanism is similar to human reasoning which is an evaluative process converging to the most satisfactory result. The main idea arises from the consistent labeling problem in computer vision. This method is originally applied to scene analysis of line drawings. Later, it is applied to matching, constraint satisfaction and multi sensor fusion by several authors [8], [16] (and see references cited there). Recently, this method is used in knowledge aggregation by Landy and Hummel [9].Comment: Appears in Proceedings of the Second Conference on Uncertainty in Artificial Intelligence (UAI1986

Massive gauge field theory without Higgs mechanism

It is argued that the massive gauge field theory without the Higgs mechanism can well be set up on the gauge-invariance principle based on the viewpoint that a massive gauge field must be viewed as a constrained system and the Lorentz condition, as a constraint, must be introduced from the beginning and imposed on the Yang-Mills Lagrangian. The quantum theory for the massive gauge fieldis may perfectly be established by the quantization performed in the Hamiltonian or the Lagrangian path-integral formalism by means of the Lagrange undetermined multiplier method and shows good renormalizability and unitarity